[EM] Margins Sorted Top-Ratings

[Ted Stern]

On Sat, Jan 27, 2024 at 2:36 PM C.Benham <cbenham at adam.com.au> wrote:

>
> I think  Margins Sorted Top Ratings would be a good  relatively burial
> resistant public proposal.
>
> * Voters rank from the top however many candidates they wish.
> Equal-ranking allowed.
>
> Give each candidate a score according to the number of ballots on which
> they are ranked below
> no others.
>
> Line them up in that order, highest to lowest.
>
> Check to see if all the candidates above bottom in this order pairwise
> beat the candidate immediately
> below them.
>
> If they do then elect the candidate highest in the order.
>
> If not begin with the pair that is pairwise out of order by the highest
> margin and swap them.
>

This is exactly the opposite of the rule for other Sorted Margins methods.
In Approval Sorted Margins (see electowiki), the pairwise out of order pair
with *minimum* margin is the pair that is swapped. The intent here is to
make the minimal change to the seed ranking in order to get pairwise
ordering.

Did you mean to introduce a new sorted margins method? I think swapping the
*maximum* margin pair would have very odd behavior.


> (if there is an exact tie in the size of the margin then swap the
> tied-margin pair lowest in the order).
>
> Repeat until no pair of adjacent candidates are pairwise out of order
> and then elect the highest-ordered
> candidate. *
>
> This could also use ratings ballots.
>
> This meets Condorcet, but can be at least be explained (if not sold)
> without reference to Condorcet or Smith.
>
> It would be as monotonic as it is possible for a Condorcet method to be.
>
> For the sake of simplicity (and elegance) it has some short-comings.
> When there is a top cycle, voters who
> didn't top-rate (rank below no other candidates) any of the candidates
> in the Smith set are disadvantaged by comparison
> those that did.  It would also fail Clone-Independence.
>
> A much more complicated method idea I had (that would be the same thing
> with three candidates):
>
> *Voters rank from the top however many candidates they wish.
> Equal-ranking allowed.
>
> (1) Eliminate (drop from the ballots and henceforth ignore) all
> candidates not in the Smith set.
>
> (2) Score the remaining candidates according to their minimum pairwise
> scores, with ballots that rank two candidates
> equal-top contributing a whole vote to each of the two candidate's
> scores against each other. Otherwise ballots that
> rank two candidates equal below top contribute zero to their pairwise
> scores against each other.
>

So, a variation on equal-rated-whole, with only equal-top having erw,
otherwise equal-rated-zero.

It appears that your tied-at-top-erw array must be recounted after the
initial elimination down to the Smith set, so it's not summable on first
pass tabulation, correct?


>
> (A possible variation is that they contribute half a vote to each if
> they are ranked below top and above bottom.)
>
> (3) Eliminate all candidates that don't have a "short" (one or two
> steps) beatpath to every candidate with a higher minimum
> pairwise score.
>

I understand the motivation, but it seems a bit complex for a public
proposal.


>
> (4).  Repeat step 2.  Then margins-sort the resulting scores and elect
> the highest-ordered candidate.*
>

So, another re-tabulation, which again loses initial summability.

This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma,
> Non-Drastic Defense.
>
> Chris Benham
>
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